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Theorem curry1val 7471
Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
curry1.1 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
Assertion
Ref Expression
curry1val ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 curry1.1 . . . 4 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
21curry1 7470 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
32fveq1d 6376 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4 eqid 2764 . . . . . . . . . 10 (𝑥𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥))
54dmmptss 5816 . . . . . . . . 9 dom (𝑥𝐵 ↦ (𝐶𝐹𝑥)) ⊆ 𝐵
65sseli 3756 . . . . . . . 8 (𝐷 ∈ dom (𝑥𝐵 ↦ (𝐶𝐹𝑥)) → 𝐷𝐵)
76con3i 151 . . . . . . 7 𝐷𝐵 → ¬ 𝐷 ∈ dom (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
8 ndmfv 6404 . . . . . . 7 𝐷 ∈ dom (𝑥𝐵 ↦ (𝐶𝐹𝑥)) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
97, 8syl 17 . . . . . 6 𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
109adantl 473 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
11 fndm 6167 . . . . . . 7 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
1211adantr 472 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom 𝐹 = (𝐴 × 𝐵))
13 simpr 477 . . . . . . 7 ((𝐶𝐴𝐷𝐵) → 𝐷𝐵)
1413con3i 151 . . . . . 6 𝐷𝐵 → ¬ (𝐶𝐴𝐷𝐵))
15 ndmovg 7014 . . . . . 6 ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶𝐴𝐷𝐵)) → (𝐶𝐹𝐷) = ∅)
1612, 14, 15syl2an 589 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → (𝐶𝐹𝐷) = ∅)
1710, 16eqtr4d 2801 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ ¬ 𝐷𝐵) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
1817ex 401 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (¬ 𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
19 oveq2 6849 . . . 4 (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
20 ovex 6873 . . . 4 (𝐶𝐹𝐷) ∈ V
2119, 4, 20fvmpt 6470 . . 3 (𝐷𝐵 → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
2218, 21pm2.61d2 173 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((𝑥𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
233, 22eqtrd 2798 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3349  c0 4078  {csn 4333  cmpt 4887   × cxp 5274  ccnv 5275  dom cdm 5276  cres 5278  ccom 5280   Fn wfn 6062  cfv 6067  (class class class)co 6841  2nd c2nd 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-1st 7365  df-2nd 7366
This theorem is referenced by:  nvinvfval  27885  hhssabloilem  28508
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